Deriving Category Probabilities From Cumulative Logits in Multiple Imputation

From Cumulative Probabilities to Category Probabilities

A proportional-odds model describes an ordinal response through probabilities of being at or below successive category thresholds. But an imputation needs a probability for each possible category, such as “mild,” “moderate,” or “severe.” The conversion is simple once the cumulative probabilities are understood: the probability of one category is the difference between two neighboring cumulative probabilities. The resulting category probabilities determine the distribution from which a missing ordinal value is drawn.

For example, if \(Y\) has four ordered categories, the model may provide \(P(Y\leq 1\mid X=x)\), \(P(Y\leq 2\mid X=x)\), and \(P(Y\leq 3\mid X=x)\). These are not the probabilities of categories 1, 2, and 3 individually. Each cumulative probability includes all categories up to its threshold.

Model and Conditions for the Conversion

Use the following proportional-odds parameterization, consistent with the cumulative-logit model for ordinal imputation. For cutpoints \(c_1<\cdots<c_{K-1}\), predictor vector \(x\), and regression coefficient vector \(\beta\), the model is

Here \(\operatorname{logit}(q)=\log\{q/(1-q)\}\), and \(\operatorname{expit}(z)=1/(1+e^{-z})\) is its inverse. Thus \(F_k(x)=\operatorname{expit}(c_k-x^\mathsf{T}\beta)\). The sign convention matters: with this form, increasing \(x^\mathsf{T}\beta\) shifts the cumulative probabilities downward. Other texts may write an algebraically different but equivalent sign convention.

The conversion to category probabilities relies on the cumulative probabilities being ordered. Since the expit function is increasing, \(c_1<\cdots<c_{K-1}\) implies \(F_1(x)<\cdots<F_{K-1}(x)\) at every fixed \(x\). Together with \(F_0(x)=0\) and \(F_K(x)=1\), this guarantees \(p_k(x)\geq 0\) and that the probabilities sum to one. The model also presumes the categories have a meaningful order and that the conditional distribution of the response is adequately represented by the specified cumulative-logit model and predictors. In MICE, draws are made conditional on the predictors available for the record; the usual missing-data conditions for valid multiple imputation must also be appropriate to the analysis.

Why the Differences Form a Probability Distribution

The conversion follows directly from the events defining the cumulative probabilities. For \(k\geq 2\), the event \(Y\leq k\) consists of the disjoint events \(Y\leq k-1\) and \(Y=k\). Therefore,

For the first category, \(P(Y=1\mid X=x)=F_1(x)\), since category 1 is the only category at or below 1. For the last category, \(P(Y=K\mid X=x)=1-F_{K-1}(x)\), because it is the complement of \(Y\leq K-1\). Summing all category probabilities gives

Nonnegativity follows from the ordering \(0=F_0(x)\leq F_1(x)\leq\cdots\leq F_{K-1}(x)\leq F_K(x)=1\). Thus the differences are a valid categorical distribution. This argument is not merely a numerical check: it explains why the cumulative probabilities must be coherent before they can be converted into category probabilities.

Drawing an Imputed Ordinal Value

Once category probabilities have been calculated, an imputation requires a random category draw, not just a list of probabilities. One convenient method is inverse-CDF sampling. Generate \(U\sim\operatorname{Uniform}(0,1)\), independently for the record conditional on the fitted model draw, and assign the smallest category \(k\) for which \(U\leq F_k(x)\). Set \(F_K(x)=1\) so the final category is always covered. Equivalently, category \(k\) is selected when \(F_{k-1}(x)<U\leq F_k(x)\).

Checks, Alternatives, and Failure Modes

Several checks make this calculation easier to audit. At each predictor profile, verify that cumulative probabilities are between zero and one and are nondecreasing. Verify that each differenced probability is nonnegative and that the unrounded category probabilities sum to one, up to numerical precision. For a simulated imputation, check that every drawn value is one of the allowed ordinal categories and that the random-number intervals correspond to the computed cumulative boundaries.

If cumulative probabilities are out of order, differencing them can produce a negative “probability.” Do not repair this by taking absolute values or silently truncating negative differences: that changes the model output and can conceal a model-fitting, parameterization, or computation error. Recheck the cutpoint order, the sign convention, and the implementation of the cumulative-logit formula. With the stated proportional-odds model and ordered cutpoints, cumulative probabilities should be ordered for every predictor profile.

A common mistake is to treat \(F_2(x)=P(Y\leq2\mid X=x)\) as \(P(Y=2\mid X=x)\). In the numerical example, \(F_2=0.3543\), but \(P(Y=2\mid X=x)=F_2-F_1=0.1864\). The cumulative probability includes category 1 as well as category 2. Another common mistake is to round cumulative values first and then subtract them. Carry full precision through the subtractions, and round only the displayed category probabilities.

A deterministic alternative is to impute the category with the largest probability. In the four-category example, that would always assign category 4 because its probability, \(0.3775\), is the largest. This is not equivalent to a stochastic draw: it understates imputation variability and produces no category 1, 2, or 3 values for records with this same profile. Drawing from the category distribution preserves the model-implied uncertainty in the missing response. A further alternative is to use another model for the ordinal response, but the probability-conversion step taught here applies specifically to cumulative probabilities from the stated cumulative-logit model.

The calculation itself does not establish that the model is well specified. Poorly chosen predictors, inadequate information for a response pattern, or a mismatch between the fitted model and the conditional distribution can still yield misleading imputations. Inspect the fitted model and the imputed distributions in context; check that categories are represented plausibly across relevant predictor patterns and that the generated values respect the response’s ordering and coding. These checks assess the implementation and plausibility, but they do not prove that missingness assumptions or the model are correct.